Homoclinic Points Cr-Created under Hypotheses by Probability Measures

Nobuo Aoki and Masatoshi Oka
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Nobuo Aoki: Tokyo Metropolitan University, Department of Mathemtics
Masatoshi Oka: Science University of Tokyo, Department of Mathematics

A chapter in Probability Measures on Groups X, 1991, pp 1-9 from Springer

Abstract: Abstract Let M be a closed manifold and f: M → M be a Cr diffeomor-phism. Let p ∈ M be a hyperbolic fixed point of f. Then the stable and unstable sets of p are denoted respectively by % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabiqaaa % qaaiaadEfadaahaaWcbeqaaiaadofaaaGcdaqadaqaaiaadchaaiaa % wIcacaGLPaaacqGH9aqpdaGadaqaaiaadIhadaabbaqaamaaxababa % GaciiBaiaacMgacaGGTbaaleaacaWGUbGaeyOKH4QaeyOhIukabeaa % aOGaay5bSdGaamizamaabmaabaGaamOzamaaCaaaleqabaGaamOBaa % aakmaabmaabaGaamiEaaGaayjkaiaawMcaaiaacYcacaWGWbaacaGL % OaGaayzkaaGaeyypa0JaaGimaaGaay5Eaiaaw2haaaqaaiaadEfada % ahaaWcbeqaaiaadwhaaaGcdaqadaqaaiaadchaaiaawIcacaGLPaaa % cqGH9aqpdaGadaqaaiaadIhadaWfqaqaamaaeeaabaGaciiBaiaacM % gacaGGTbaacaGLhWoaaSqaaiaad6gacqGHsgIRcqGHEisPaeqaaOGa % amizamaabmaabaGaamOzamaaCaaaleqabaGaeyOeI0IaamOBaaaakm % aabmaabaGaamiEaaGaayjkaiaawMcaaiaacYcacaWGWbaacaGLOaGa % ayzkaaGaeyypa0JaaGimaaGaay5Eaiaaw2haaaaaaaa!6F1C! $$ \begin{array}{*{20}{c}} {{W^S}\left( p \right) = \left\{ {x\left| {\mathop {\lim }\limits_{n \to \infty } } \right.d\left( {{f^n}\left( x \right),p} \right) = 0} \right\}} \\ {{W^u}\left( p \right) = \left\{ {x\mathop {\left| {\lim } \right.}\limits_{n \to \infty } d\left( {{f^{ - n}}\left( x \right),p} \right) = 0} \right\}} \\ \end{array} $$ which are Cr injectively immersed submanifolds of M. The points of intersection of the closure of Ws(p) with Wu(p) or of the closure of Wu(p) with Ws(p), different from p, is called almost homoclinic points to p.

Keywords: Probability Measure; Unstable Manifold; Closed Manifold; Orbit Structure; Homoclinic Point (search for similar items in EconPapers)
Date: 1991
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DOI: 10.1007/978-1-4899-2364-6_1

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